Solve the problem The number of mosquitos Mx in millions in

Solve the problem

The number of mosquitos M(x), in millions, in a certain area depends on the June rainfall x, in inches: M(x) = 18x-x^2. What rainfall produces the maximum number of mosquitoes?

Solution

Note that M\'(x) = 0 determines the relative extrema of M(x).

M\'(x) = 18 - 2x
0 = 18 - 2x
18 = 2x
x = 9

Then,

M\'\'(x) = -2 . . This is second derivative test

Since M\'\'(x) < 0 for all x values, this yields the maximum at x = 9. Check x = 9 for M(x)!

M(9) = 18(9) - (9)²
==> 162 - 81

  ==> 81

Hence, the max. amount of mosquitoes is 81 million.

I hope this helps!